The sublattice polarization of the narrow bands is defined via the eigenvalues of the \(2\times 2\)  projected sublattice matrix \(\mathcal {S}_{ij}(k) = \langle \Psi _i(k) | \sigma _z | \Psi _j(k) \rangle \) , where \(\Psi _{i}(k)\)  is the Bloch state with the momentum of \(k\)  in the band \(i\) , and \(\sigma _z\)  is the sublattice polarization operator. Because of the \(C_2\mathcal {T}\)  symmetry, the two eigenvalues of the projected \(\sigma _z\)  have the same magnitude with opposite signs, \(\pm 1\)  corresponding to the perfect polarization obtained in the chiral limit [1]}. The sublattice polarization calculated based on the \(H_{eff}^{\mathbf {K}}\)  is shown in the Fig. REF  for the two microscopic models. While the narrow band Hilbert space in either model is not perfectly polarized, the one proposed in Ref. [2]} has larger sublattice polarization than the one in Ref. [3]}, implying the former model is closer to the chiral limit than the latter.
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